Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| socle | C22×C53 | C22×D4 |
| prime radical | C53×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#533 | C2 | semidirect |
| 3. C53×#491 | C2 | semidirect |
| 4. C53×#490 | C2 | semidirect |
| 5. C53×#489 | C2 | semidirect |
| 6. C53×#416 | C2 | semidirect |
| 7. C2×#97×C53 | C2 | semidirect |
| 74. #3135 | C53 | direct (N × H) |
| 79. #533 | C2×C53 | semidirect |
| 81. #491 | C2×C53 | semidirect |
| 82. #490 | C2×C53 | semidirect |
| 83. #489 | C2×C53 | semidirect |
| 84. #416 | C2×C53 | semidirect |
| 85. C2×#97 | C2×C53 | semidirect |
| 86. C53 | #3135 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#533 | 2 |
| C53×#532 | 2 |
| C53×#491 | 2 |
| C53×#490 | 2 |
| C53×#489 | 2 |
| C53×#416 | 2 |
| C2×#97×C53 | 2 |
| #3135 | 53 |